The present invention relates to a design method, an optimization analyzing apparatus and a storage medium having a stored optimization analyzing program for a multi-component material and more particularly, to a design method, an optimization analyzing apparatus and a storage medium having a stored optimization analyzing program for design of a multi-component material composed of many components, for example design of a rubber compound for a tire.
Material design is to determine compositional ratios of components for a material to obtain the material having target mechanical behaviors, production conditions therefor and the like. The mechanical behaviors include physical quantities, such as physical properties of a material itself, sizes thereof and the like.
A conventional material design has mainly been conducted through experiences or trial and error and there has been difficulty in designing a material in the case of a material composed of three or more components. For example, in designing a rubber member for producing a tire, since various physical properties are considered, there have generally been taken procedures that a plurality of component materials having known physical properties are selected beforehand, the rubber member including predetermined compositions of respective component materials are as trial produced for tests and such tests are repeated until the member showing target performances in terms of a Young""s modulus and the like is achieved. In this manner, the rubber member is designed and developed.
There has been proposed a material design method in which compositional ratio of multi-component materials can automatically be determined with use of a hierarchical neural network, if material properties are designated (see xe2x80x9cDesign Method for Multi-component Material Using Neural Networkxe2x80x9d in the papers presented at the first symposium on optimization sponsored by Japan Society of Mechanical Engineers, pp. 57-62).
In this method, however, if it is tried to determine compositional ratio of components by inputting material properties, which are slightly different from ones of an actual material, negative compositional ratio of the material is output, that is, compositional ratio of components with which a material cannot actually be composed is obtained and thus there arises a case where obtained compositional ratio cannot be used. Further, mechanical behaviors, such as material properties, except for mechanical behaviors which can be obtained the target performances cannot meet requirements from the market, so that the method has in fact been unable to apply in many practical aspects.
Another method has been known in which a polynominal is assumed for correlating compositional ratios of multi-component materials composed of a plurality of components and production conditions with mechanical behaviors of the materials and a calculating means determining coefficients of the polynominal by method of least squares and optimization are combined to each other (see authored by Manabu Iwasaki, xe2x80x9cPlanning and Analysis in Mixed Experimentsxe2x80x9d published by Scientist Co.,).
In this method, however, when optionality enters in assuming the polynominal or the number of the multi-component materials is three or more, the assumption of the polyniminal becomes difficult and the correlating cannot be performed with a high accuracy. For this reason, a plan of material design obtained has not been useful.
In view of the above facts, it is an object of the present invention to provide an optimization analyzing apparatus for a multi-component material and a design method therefor, in which a design of a material composed of a plurality of components is facilitated and since optimization is conducted while constraint conditions are considered, a design range of compositional ratios of components of multi-component materials and a desired range of a mechanical behavior can in advance be set.
The present inventor has studied various aspects in order to achieve the above mentioned object and as a result, has paid attention to application of a non-linear prediction technique, for example, a neural network, in which a neural circuit network of a higher animal is modeled in an engineering manner, and optimization design approach, both the technique and the approach being utilizing in some fields except for a material design field, to a special field of material design, carried out a research and established a design method in which mechanical behaviors, such as a Young""s modulus, are considered in a concrete manner.
A design method for a multi-component material of the present invention comprises the steps of: (a) determining a conversion system in which a non-linear correspondence between compositional ratios of multi-component materials composed of a plurality of components and mechanical behaviors of the multi-component materials is established; (b) determining an objective function expressing the mechanical behaviors and setting a constraint condition constraining an allowable range of at least one of the mechanical behaviors and the compositional ratios of the multi-component materials; and (c) determining a compositional ratio of the multi-component materials which gives an optimal solution of the objective function on the basis of the objective function and the constraint condition by using the conversion system determined in the step (a) to design a multi-component material based upon the compositional ratio of the multi-component materials determined.
The mechanical behaviors of the multi-component materials composed of the plurality of components, such as Young""s modulus, tan xcex4 and the like in a rubber member, are determined by the compositional ratios of components thereof. However, there are many cases where the mechanical behaviors are not changed in a linear manner, even though the compositional ratios are changed linearly. Therefore, in the step (a) of the present invention, the conversion system establishing a correspondence between the compositional ratios and the mechanical behaviors including a non-linear correspondence therebetween in advance is determined in advance. This conversion system can be determined with use of a non-linear prediction technique in which such a neural circuit network as neural network is modeled by an engineering approach.
In the step (b), the objective function expressing the mechanical behaviors is determined and the constraint condition constraining the allowable range of at least one of the mechanical behaviors and the compositional ratios of the multi-component materials is determined. As the objective function expressing mechanical behaviors, for example, a physical quantity governing excellency of a rubber member, such as a Young""s modulus, tan xcex4 and the like can be used. As the constraint constraining the allowable range of at least one of the mechanical behaviors and the compositional ratio of the multi-component materials, there are in a rubber member, for example, constraints for a Young""s modulus and Poison""s ratio of the rubber member and a constraint for a mass of at least one component of the multi-component materials. The objective function, design variable and constraint condition are not limited to the above mentioned but various kinds can be determined according to a design object of a tire.
In the step (c), the compositional ratio of the multi-component materials which give an optimal solution of the objective function is determined based on the objective function and the constraint condition by using the conversion system determined in the step (a) and then the multi-component material is designed on the basis of the compositional ratio of the multi-component materials thus determined. In this manner, the conversion system is determined so as to establish the non-linear correspondence between the compositional ratios of components of the multi-component materials composed of a plurality of components and the mechanical behaviors of the multi-component materials and then according to the conversion system a mutual relation is found between the compositional ratios of a plurality of components included in the materials and the mechanical behaviors thereof. Therefore, a multi-component material can be designed with a high precision and a lesser optionality by designing a multi-component material based on compositional ratios of the multi-component materials after the compositional ratio which give an optimal solution of the objective function is obtained. In the step (c), a value(s) of a design variable(s) which gives an optimal solution of the objective function can be obtained while the constraint condition is considered.
In the case where a multi-component material is designed in the step (c) , the compositional ratio of the multi-component materials is set as a design variable, the value of the design variable which gives and optimal solution of the objective function is determined by using the conversion system determined in the step (a) while considering the constraint condition, and the multi-component material can be designed based on the design variable which gives the optimal solution of the objective function. In such a manner, if the constraint condition is considered, the allowable range of at least one of the mechanical behaviors and the compositional ratio of the multi-component materials can be considered and thereby not only can a range of a material design be defined in advance but a desired range thereof can also be set.
In the case where the value of the design variable is obtained in the step (c), a changed quantity of the design variable which gives the optimal solution of the objective function is predicted, while considering the constraint condition, based on a sensitivity of the objective function which is a ratio of a changed quantity of the objective function to a unit change quantity of the design variable and a sensitivity of the constraint condition which is a ratio of a changed quantity of the constraint condition to a unit change quantity in the design variable, and a value of the objective function obtained when the design variable is changed in a corresponding manner to a predicted quantity and a value of the constraint condition obtained when the design variable is changed in a corresponding manner to a predicted quantity are calculated so that a value of the design variable which gives the optimal solution of the objective function is effectively obtained by using the conversion system determined in the step (a) based on the predicted and calculated values while considering the constraint condition. Thereby, the value of the design variable which gives the optimal solution of the objective function under consideration of the constraint condition can be obtained. Then, a multi-component material can be designed by modifying compositional ratio or the like based on the design variable which gives the optimal solution of the objective function.
It has been known that to obtain an optimal solution in a general optimization approach is analogous to climbing a mountain. In this case, the optimal solution corresponds to the peak of the mountain if a height of the mountain is related with a performance or the like. Therefore, when the objective function is simple, since a design space is a shape like a mountain as shown on FIG. 8, the optimal solution can be obtained by an optimization approach which is based on a mathematical programming. Description will roughly be given on a design of a multi-component material using a typical drawing in FIG. 8 as a model, in which mountain climbing is used for illustration of optimization. The conversion system defines a non-linear correspondence between compositional ratios of the multi-component materials and mechanical behaviors of the multi-component materials. The conversion system is shown to be at a level (as a contour) in the design space (in the shape like a mountain). That is, there is a case where mechanical behaviors of multi-component materials are correlated with a plurality of compositional ratios of the multi-component materials and generally as the mechanical behavior approaches an optimal solution, the ranges of the compositional ratios are smaller as is a contour. The ranges of the compositional ratios of the multi-component materials are, in general limited by constraints in design and an actually allowable range, so that a relation between the mechanical behaviors and the compositional ratios of the multi-component materials can be restrained by a fence along a ridge of the mountain, as shown in FIG. 8. If the fence is assumed as constraint conditions, the relationship is considered to climb the mountain as shown in FIG. 8 with the help of an optimization approach, such as a mathematical programming or the like up to the peak of the mountain where an optimal solution can be obtained for the objective function in such a manner that the relationship is kept from going over to the outside of the fence by changing the design variable within the conversion system.
Besides, the constraint conditions (the fence) are effective for a guide in climbing a mountain in an optimization approach in addition to setting desired ranges of design of the compositional ratios of the multi-component materials and of the mechanical behaviors. That is, without a constraint condition, not only is a time for calculation increased but the calculation is not converged and furthermore the calculation is departed from the desired design range of the compositional ratio of the multi-component materials and the desired range of the mechanical behaviors.
In the present invention, when an optimal solution is obtained by the steps (a) to (c), execution of the following steps of (d) to (f) is indispensable to obtain the optimal solution. In more detail, the step (c) can comprises the steps of: (d) selecting as a design variable one compositional ratio of the multi-component materials included in the conversion system determined in the step (a); (e) changing a value of the design variable selected in the conversion system determined in the step (a) until an optimal solution of the objective function is given by using the conversion system determined in the step (a) while considering the constraint condition; and (f) designing the multi-component material based on the compositional ratio of the multi-component materials according to the design variable which gives the optimal solution of the objective function. In the step (d), the one compositional ratio of the multi-component materials included in the conversion system in the step (a) is selected as the design variable. In the next step (e), the design value to be selected in the conversion system is changed until an optimal solution of the objective function is given by using the conversion system determined in the step (a) while considering the constraint condition. Thereby, the design value changes subtly or gradually to obtain a optimal solution of the objective function. In the step (f), the multi-component material is designed based on the compositional ratio according to the design variable which gives an optimal solution of the objective function. In such a manner, since the one compositional ratio of the multi-component materials included in a conversion is selected as the design variable and the design variable to be selected is changed in the conversion system while considering the constraint condition until an optimal solution of the objective function is obtained, without beforehand preparation for a value of the design variable which gives the optimal solution of the objective function, a design value, which is close to a desired value of the design variable, may be selected in the conversion system and thereby a design of the multi-component material can be practiced with a high precision and a lesser optionality.
In this case, in the step (b), the constraint condition, in which an allowable range of at least one of the mechanical behaviors other than the determined objective function and the compositional ratios of the multi-component materials is constrained can be determined. In such a manner, the mechanical behaviors other than the objective function as a constraining allowable range can be used by determining the constraint condition, in which an allowable range of at least one of the mechanical behaviors other than the determined objective function and the compositional ratios of the multi-component materials is constrained.
In the step (e), a change in the design variable which gives an optimal solution of the objective function is predicted while considering the constraint condition based on a sensitivity of the objective function which is a ratio of a changed quantity of the objective function to a unit change quantity of the design variable and a sensitivity of the constraint which is a ratio of a changed quantity of the constraint condition to a unit change quantity of the design variable, and a value of the objective function obtained when the design variable is changed in a corresponding manner to a predicted quantity and a value of the constraint condition obtained when the design variable is changed in a corresponding manner to a predicted quantity are calculated. Besides, a value of the design variable can be changed until the value of the design variable gives an optimal solution of the objective function by using the conversion system determined in the step (a) based on the predicted and calculated values while considering the constraint condition. Thereby a value of the design variable can be obtained with ease during an optimal solution of the objective function is given by calculating a value of the objective function obtained when a value of a design variable is changed in a corresponding manner to a predicted quantity and a value of the constraint obtained when a value of a design variable is changed in a corresponding manner to a predicted quantity.
The present inventors has established a definite design method for a multi-component material after various studies by achieving an idea that a genetic algorithmic approach which is utilized in a field from that of the invention is applied to a special field of material design.
According to a design method of a multi-component material of the present invention, the step (c) comprises the steps of: define the compositional ratios of the multi-component materials in the conversion system determined in the step (a) as material base models to determine groups for selection including a plurality of material base models; determining the objective function, a design variable, a constraint and an adaptive function which can be evaluated from the objective function, for each material base model of the groups for selection; selecting two material base models from the groups for selection; conducting at least one of producing new material base models by cross overing design variables of the two material base models, at a predetermined probability with each other, and producing new material base models by modifying in part the design variables of at least one of the two material base models; changing the design variables of the new material base models produced to obtain an objective function, a constraint and an adaptive function of the new material base models produced by using the conversion system determined in the step (a); storing the material base models whose design variables have been changed and the material base models whose design variables have not been changed; repeating the storing steps until the number of the stored material base models reaches a predetermined number; determining whether or not new groups comprising the stored material base models of the predetermined number satisfies a predetermined convergence condition; wherein if not, the new groups are defined as the groups for selection and the above steps are repeated until the new groups satisfy the predetermined convergence condition; and if the predetermined convergence condition is satisfied, designing a multi-component material based on the compositional ratio of the multi-component materials obtained by the design variable which gives an optimal solution of the objective function from the predetermined number of the stored material base models by using the conversion system determined in the step (a) while considering the constraint condition.
In the step (a) the conversion system can be constructed with data in a multi-layered feed forward type neural network which has learned so as to convert the compositional ratios of the multi-component materials to the mechanical behaviors thereof.
As mentioned above, there are: a mathematical programming, a genetic algorithm and the like in a general optimization approach, and obtaining an optimal solution is understood to be analogous to climbing a mountain. At this point, since a height of the mountain is related to a performance or the like, the optimal solution corresponds to the peak of the mountain. In the case where an objective function is simple, a design space thereof (a mountain shape) is like Mt. Fuji having one peak as shown in FIG. 8, the optimal solution can be obtained an optimization approach based on a mathematical programming. However, when an objective function is more complex, a design space has a plurality of peaks, as shown in FIG. 9, an optimal solution cannot be obtained by the optimization approach based on a mathematical programming. The reason is that the optimization approach based on a mathematical programming recognizes a peak which is first reached by chance as an optimum solution among the plurality of peaks. A genetic algorithm has been proposed in order to solve this problem, but it requires tremendous amounts of experiments and computational time and sometimes calculation has not been converged. A neural network which can be used in the step (a) can be expected to have a prediction and a decision, both with higher precision than a linear transformation multi-variable analysis, a learning of a correlating a plurality of input data can be effected and thereby any function can be converted to approximation with any precision if the number of units in an intermediate layer is increased and besides the analysis has an advantage that it is excellent in extrapolation (see a book authored by Hideki Toyota xe2x80x9cNon-Linear Multi-Variate Analysisxe2x80x94Approach by Neural Networkxe2x80x9d p. 11 to 13, p. 162 to 166, published by Asakura Book Store in 1996). This conversion system can be determined by use of a non-linear prediction technique in which a neural circuit network such as a neural network is modeled in an engineering way.
An optimal solution can be obtained by applying the neural network in a combination with the above mentioned optimization approach in a limited time, even when the objective function becomes complex.
In the case where design and development are conducted based on a design method of the present invention, it is made possible to conduct jobs of a design of a multi-component material having best mechanical behaviors to a performance evaluation of the multi-component material, mechanical behaviors can be achieved from in a mainly by a computer calculation which is different from a conventional design development in which trial and error are fundamental. Therefore conspicuous increase in efficiency is achieved and development costs can be decreased.
If a rubber compound is formed on the basis of a compositional ratio of multi-component materials designed by according to the above mentioned design method of a multi-component material, which mean that the rubber compound formed is constituted from respective components of multi-component materials having best mechanical behaviors. Accordingly, such a mixing contents as a quantity of carbon (% by weight), a quantity of a rubber chemical (% by weight) and the like can directly be determined according to applied conditions such as a production condition and a cost.
The above design method for a multi-component material can be realized by an optimization analyzing apparatus comprising; a conversion system calculating means for obtaining a non-linear corresponding relation between compositional ratios of multi-component materials composed of a plurality of components and mechanical behaviors of the multi-component materials; input means for inputting an objective function and a constraint condition as optimization items by determining the objective function expressing the mechanical behaviors and determining the constraint condition which constrains an allowable range of at least one of the mechanical behaviors and the compositional ratios of the multi-component materials; and optimization calculation means for obtaining a compositional ratio of the multi-component materials which gives an optimal solution of the objective function based on the optimization items inputted from the input means by using the conversion system calculating means.
The non-linear corresponding relation between, on the one hand, the compositional ratios of the multi-component materials and a condition to be applied to the multi-component material and, on the one hand, the mechanical behaviors of the multi-component materials can be obtained by the conversion system calculation means.
The optimization calculation means can comprises: selecting means for selecting one compositional ratio of the compositional ratios of the multi-component materials included in the conversion system calculation means as a design variable; changing means for changing a value of the design variable selected from the conversion calculation means until the optimal solution of the objective function gives the optimal solution, while considering the constraint condition; optimal solution calculation means for calculating the value of the design variable until the optimal solution of the objective function is given by using the conversion system calculating means; and design means for designing a multi-component material based on the compositional ratio at the design variable which gives the optimal solution of the objective function.
The optimization calculating means is constructed as to effect the steps of: defining the compositional ratios of the multi-component materials in the conversion system determined in the conversion system calculation means as material base models to determine groups for selection composed of a plurality of material base models; for each of the material base models in the groups for selection, determining the objective function, the design variable, the constraint and an adaptive function which can be evaluated from the objective function; selecting two material base models from the groups; effecting at least one of producing new material base models by cross overing the design variables of the two material base models, at a predetermined probability with each other and producing new material base models by modifying in part the design variables at least one of the two material base models; obtain an objective function, a constraint condition and an adaptive function of the new material base models using the conversion system determined in the conversion calculation means by changing the design variables of new material base models; storing the material base models whose design variables have been changed and the material base models whose design variables have not been changed; repeating the storing steps until the number of the stored material base models reaches a predetermined number; determining whether or not new group comprising the stored material base models of the predetermined number satisfies a predetermined convergence condition; wherein if not, the new groups are defined as the groups for selection and the above steps are repeated until the new groups satisfy the predetermined convergence condition; and if the predetermined convergence condition is satisfied, designing a multi-component material based on the compositional ratio of the multi-component materials obtained from the design variable which gives the optimal solution of the objective function of one of the predetermined number of the stored material base models by using the conversion system determined in the conversion system calculation means while considering the constraint.
The conversion system calculation means comprises a multi-layered feed forward type neural network which has learned so as to convert the compositional ratios of the multi-component materials to the mechanical behaviors thereof.
The above mentioned design method of a multi-component material can provide a storage medium having a stored optimization analyzing program for a multi-component material. The storage medium includes a program according to the following procedures and is portable.
The storage medium having the stored optimization analyzing program for a multi-component material is a storage medium having a stored optimization analyzing program for design of a multi-component material by a computer. The optimization analyzing program is programmed to determine a non-linear corresponding relation between compositional ratios of multi-component materials and mechanical behaviors of the multi-component materials, to determine an objective function expressing the mechanical behaviors, and to determine a constrain constraining an allowable range of at least one of the mechanical behaviors and the compositional ratios of the multi-component materials and determine one of the compositional ratios of the multi-component material, which gives an optimal solution of the objective function, based on the corresponding relation, the objective function and the constraint to design the multi-component material based on the one of the compositional ratios.
The design of a multi-component material based on the compositional ratios of the multi-component materials conducts the steps of: selecting as a design variable one of the compositional ratios of the multi-component materials included in the determined corresponding relation based on the determined corresponding relation, the objective function, and the constraint; changing a value of the design variable selected from the determined corresponding relation until an optimal solution of the objective function is given while considering the constraint condition; and designing the multi-component material based on the compositional ratio of the multi-component materials obtained by the design variable which gives the optimal solution of the objective function.
The constraint condition can constrain an allowable range of at least one of the mechanical behaviors other than the determined objective function and the compositional ratios of the multi-component materials.
The change in the design variable which gives the optimal solution of the objective function is predicted while considering the constraint based on a sensitivity of the objective function which is a ratio of a changed quantity of the objective function to a unit change quantity of the design variable and a sensitivity of the constraint which is a ratio of a changed quantity of the constraint to a unit change quantity in the design variable, and a value of the objective function when the design variable is changed in a corresponding manner to a predicted quantity and a value of the constraint when the design variable is changed in a corresponding manner to the predicted quantity are calculated. Besides, a selected value of the design variable can be changed based on the predicted and calculated values while considering the constraint until the optimal solution of the objective function is given.
A design of a multi-component material based on the compositional ratios comprises the steps of defining the compositional ratios of the multi-component materials in the determined corresponding relation as material base models determine groups for selection composed of a plurality of material base models; for the material base models in the groups for selection, determining the objective function, the design variable, the constraint and an adaptive function which can be evaluated from the objective function; selecting two material base model from the groups, effecting at least one of producing new material base models by cross overing the design variables of the two material base models, at a predetermined probability with each other and producing new material base models by modifying in part the design variables of at least one of the two material base models; obtaining an objective function, a constrained condition and an adaptive function of the material base models whose design variables have been changed; storing the material base models whose the design variables have been changed and a material base models whose design variables have not been changed; repeating the storing steps until the number of the stored material base models reaches a predetermined number; deciding whether or not new groups comprising the predetermined number of the material base models stored satisfy a predetermined convergence condition; wherein if not, the above steps are repeating until the new groups satisfy the predetermined convergence condition; and if the predetermined convergence condition is satisfied, designing a multi-component material based on the compositional ratio of the multi-component materials obtained from the design variables which gives the optimal solution of the objective function in the predetermined number of the material base models stored by using the corresponding relation while considering the constraint.
As mentioned above, according to the present invention, the present invention has an effect that since a conversion system in which a non-linear correspondence between compositional ratios of multi-component materials composed of a plurality of components and mechanical behaviors of the multi-component material are correlated to each other is determined, the conversion system, in which a corresponding relation between compositional ratios of a plurality of compositions and mechanical behaviors thereof can be found out, can be obtained with high precision and lesser optionality.
The present invention has another effect that since one of a compositional ratios of multi-component materials which gives the optimal solution of an objective function by using the conversion system is obtained, the optimal design plan which is effective with a compositional ratio of multi-component materials can be achieved.
An optimal composition of a multi-component material, for example a quantity of carbon (% by weight), a rubber chemical (% by weight) and the like can directly be determined according to applied conditions such as production conditions, a cost and the like. As optimization items to be input, for example an abstract technical information such as a particle size or a particle ratio of carbon, can be used.